Phase reconstruction from two Wigner distribution projections

The reconstruction of a signal - and in particular its phase - from the distributions associated with the instantaneous power of the signal or, more general, its fractional power spectra, is an important problem in signal processing. In spite of several successful iterative algorithms for phase reconstruction from the squared modulus of the signal and its power spectrum, the development of noniterative procedures remains an attractive research topic. The fractional power spectra, i.e., the squared moduli of the fractional Fourier transform, are now a popular tool in signal processing. As it is known, the fractional power spectra are equal to the projections of the Wigner distribution of the signal. Thus, by using a tomographic approach and the inverse Radon transform, the Wigner distribution - and therefore the signal itself, up to a constant phase factor - can be reconstructed by knowing all its projections. The method is based on the rotation in the time-frequency plane of the Wigner distribution under the fractional Fourier transform. It requires the measurements of the fractional power spectra in the entire angular region, which sometimes is impossible or very cost consuming. In the paper we first present the relationship between the fractional power spectra and the ambiguity function of a signal, and we establish the connection between the instantaneous frequency in a fractional domain and the angular derivative of the fractional power spectra. We then show that the instantaneous frequency is determined by the convolution of the angular derivative of the fractional power spectra and the signum function. From this relationship we derive a new method to reconstruct the phase of the signal from only two fractional power spectra. The new method significantly reduces the need for projections measurements and calculations; it is also direct and does not use iterative procedures. We discuss the discrete version of the proposed phase retrieval method and demonstrate its efficiency on some examples.